On top of the nonregularity of the problem (REF ), the high-dimensionality of the parameter adds an
additional layer of complexity for inference.
The reason is that the estimator that minimizes the penalized loss function does not have a tractable standard limiting distribution under high dimensionality, due to the bias induced by the penalty term.
For regular models (e.g., generalized linear models), there is a growing literature on correcting the bias from the penalty for valid inference, such as [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]},
among others.
Their main idea is to firstly construct a consistent estimator of the high dimensional parameter via proper regularization, and then remove the bias (via debiasing or decorrelation) in order to develop valid inferential statistics.
While these methods enjoy great success under regular models, it remains unclear whether they can be applied to conduct valid inference in nonregular models
such as the problem we consider in this paper.
To the best of our knowledge, our work is the first one that provides valid inferential tools for nonregular models in high dimension.
